Linear operators that preserve Boolean rank of Boolean matrices

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m× k Boolean matrix B and a k × n Boolean matrix C such that A = BC. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and k for some 1 < k ? m.